I like 5 mil cord tied through the hole in the head
and leading to a runner over the head and shoulder.
With the right length of cord you
can switch hands and chop steps with out having to deal with
a wrist loop and potentially having the loop off when you
fall.

That's a great idea about the runner. Another good idea might be to tie it into the front harness loop. Perhaps not as good as a hand loop, but it would be easier to switch hand.
I think I'll get some 3/8" webbing rather than a premade leash. More versatile.

Dave,
I use a tether attached to the head of my 70 cm ax. Its length is calibrated to stabilize my hand about 6-8 cm above the spike for either chopping steps or above the head placements when facing in on steep slopes. This length is also suitable for allowing me to switch hands when reversing direction on ascents of lower angles slopes without bothering to switch the tether back and forth between uphill hands at every switchback. Some people, as one poster mentioned, attach the tether to their harness. I never liked that when I was climbing because it was one more piece of rope near my Pearabiner(locking biner) when getting ready to belay, and I'm a BIG fan of the KISS principle when it comes to rope management and climbing in general. Also, attaching the tether to your harness would make it impossible to place the ax ubove the head without having a very long tether which could be very dangerous, IMO. Never tried it when not climbing simply because I don't wear a harness unless I'm roping up. The idea of attaching it to a cross body sling is interesting but, again, for me that would apply to a technical climbing situation. Otherwise, a sling is just one more piece of gear that add weight to your kit. The OP did not indicate he was going to be climbing so I think he could get by just fine with only a tether. My 2 cents.

On low angle snow I prefer to not use a leash because it makes switching sides faster. The real advantage of a leash on low angle snow and ice is to prevent losing it if you fall into a crevasse. When I do use a leash on glaciers I use a length of 3/8 inch webbing where the end loop is big enough to easily get my gloved hand into. Do not automatically think a leash is necessary find what works for you. Thats just my 2 cents.

If you use the runner/leash method another advantage is you
can quickly holster or stick the axe between your pack or
sweater and back for a climbing move or just to use both hands
and still have it secured.

<< An elastic sling system to avoid losing tools when climbing on rock or ice. This solves the problem for climbers who love climbing without a leash. The Single Spring is attached to the harness by looping it through the tape ring over the harness itself. The small carabiner is hooked onto the hole on the axe’s head when using it to lean on when walking or through the hole on the spike when holding the shaft in traction. In both cases it is impossible to lose the axe. The system allows the climber to swap hands or tool easily without interfering with the manoeuvre thanks to it compactness and the arm can be extended to its limit thanks to its elasticity. Compact when required, long when required. >>